No 44

The Ultimate Palindrome

What is a Palindromic Number? Examples are 11, 123321 and 2002. They read the same backwards as forwards. This is the number equivalent of a Palindrome which is a word which reads the same forwards and backwards. Hannah is an example.

Pick a number, reverse its digits and add the resulting number to the original number. If the answer is not palindromic then repeat the process. For example let's start with 87:

Step 1:	87 + 78 = 165	not palindromic so...
Step 2:	165 + 561 = 726	not palindromic so...
Step 3:	726 + 627 = 1353	not palindromic so...
Step 4:	1353 + 3531 = 4884	palindromic!

You might like to explore the result of doing this process to several other small numbers. Do all the numbers that you try end up with a palindromic number?

196 is a particularly stubborn number. Nobody knows whether eventually doing this reversal and adding process with 196 yields a palindromic number. Here is a quote from an account written by John Walker on May 25th 1990. He had written a computer program in an attempt to determine the outcome for the number 196:

"For almost three years the process of reversal and addition continued. Last night, at five minutes before midnight, the program printed the message: 'Stop point reached on pass 2415836. Number contains 1000000 digits.' and exited. The built-in endpoint had been reached; after 2,415,836 reversals and additions, 196 had grown to a number of 1,000,000 digits without ever yielding a palindrome. Does it ever produce one? Still, nobody knows."

Recently computer tests have continued up to nearly 9.5 million steps yielding a 4.9 million digit number which is not yet palindromic! The search no doubt continues!!

If you have answers, discoveries, new questions etc to do with this Number Fact then...

E-mail us at madrascollege.maths@fife.gov.uk

...and we will publish them here.

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